Fat chance Mayor Bloomberg’s vast wealth will influence the competitive race to replace him in City Hall next year.

The billionaire mayor has told people that he’s almost certain not to throw his cash into the mayoral race, after leaving the possibility open several months ago, The Post has learned.

“It wouldn’t work,” the mayor said recently, recalled a source, who said Bloomberg was referring to the prospect of helping his preferred candidate through a cash gift known as an “independent expenditure” from his Independence USA super PAC.

The mayor and those close to him have said spending his fortune on the race could backfire and end up hurting his preferred candidate — who’ll likely be City Council Speaker Christine Quinn, his close political ally.

“It would be more problem than it’s worth for both the mayor and the candidate he supports,” said a source close to Bloomberg. “There would have to be an overwhelmingly compelling reason for the mayor to do an independent expenditure. And short of his legacy being threatened, the mayor’s probably unlikely to get involved through an IE.”

Other sources say the mayor simply isn’t that impressed with anyone running to succeed him, including Quinn. He recently even asked Hillary Rodham Clinton if she would be interested in running for mayor.

Bloomberg formed a super PAC in October to support candidates throughout the nation who share his beliefs on a range of issues, including gun control and bipartisanship.

But at home, his donation could backfire on a campaign.

For starters, a large contribution could turn into a liability for Quinn, who already is facing criticism from opponents Bill Thompson, John Liu and Bill de Blasio over her close relationship with Bloomberg.

Thompson came within 4.5 points of beating Bloomberg in 2009, in large part because he painted the mayor as an out-of-touch billionaire who overturned term limits to win another four years in office.

Bloomberg could spend a limitless amount on a campaign through his super PAC, but federal and local rules require that money to be disclosed. If it turns out he coordinated with the campaign, the candidate would then be subject to the city Campaign Finance Board’s strict contribution and spending limits.

When asked whether the mayor would do an independent expenditure, Deputy Mayor Howard Wolfson replied, “There isn’t a single person you would be talking to who would have a clue what he is thinking of regarding this.”